Records

by Arnold, Barry C.; Balakrishnan, Narayanaswamy; Nagaraja, Haikady N.

• ISBN13: 9780471081081
• ISBN10: 0471081086
• Edition: 1st
• Format: Hardcover
• Copyright: 1998-07-07
• Publisher: Wiley

Summary

The first and only comprehensive guide to modern record theory and its applications Although it is often thought of as a special topic in order statistics, records form a unique area, independent of the study of sample extremes. Interest in records has increased steadily over the years since Chandler formulated the theory of records in 1952. Numerous applications of them have been developed in such far-flung fields as meteorology, sports analysis, hydrology, and stock market analysis, to name just a few. And the literature on the subject currently comprises papers and journal articles numbering in the hundreds. Which is why it is so nice to have this book devoted exclusively to this lively area of statistics. Written by an exceptionally well-qualified author team, Records presents a comprehensive treatment of record theory and its applications in a variety of disciplines. With the help of a multitude of fascinating examples, Professors Arnold, Balakrishnan, and Nagaraja help readers quickly master basic and advanced record value concepts and procedures, from the classical record value model to random and multivariate record models. The book follows a rational textbook format, featuring witty and insightful chapter introductions that help smooth transitions from one topic to another and challenging chapter-end exercises, which expand on the material covered. An extensive bibliography and numerous references throughout the text specify sources for further readings on relevant topics. Records is a valuable professional resource for probabilists and statisticians, in addition to applied statisticians, meteorologists, hydrologists, market analysts, and sports analysts. It also makes an excellent primary text for courses in record theory and a supplement to order statistics courses.

Author Biography

BARRY C. ARNOLD is Professor in the Department of Statistics at University of California, Riverside. He is a Fellow of both the American Statistical Association and the Institute of Mathematical Statistics. His publications include another Wiley book, A First Course in Order Statistics. N. BALAKRISHNAN is Professor in the Department of Mathematics and Statistics at McMaster University, Canada. He is a Fellow of the American Statistical Association. His publications include other Wiley books: A First Course in Order Statistics, Continuous Univariate Distributions - 1 & 2, and Discrete Multivariate Distributions. H. N. NAGARAJA is Professor in the Department of Statistics at Ohio State University, Columbus. His publications include another Wiley book, A First Course in Order Statistics.

Table of Contents

Notations and Abbreviations	xiii	(4)
Preface	xvii
1. Introduction	1	(6)
1.1 Who Cares?	1	(1)
1.2 A Road Map	2	(5)
2. Basic Distributional Results	7	(44)
2.1 Introduction	7	(1)
2.2 Standard Record Values Processes	7	(2)
2.3 Record Values From the Classical Model	9	(11)
2.4 Record Values From Specific Distributions	20	(2)
2.4.1 Weibull Records	20	(1)
2.4.2 Power Function Distribution Records	21	(1)
2.4.3 Pareto Records	21	(1)
2.4.4 Extreme Value Records	22	(1)
2.5 Record Times and Related Statistics	22	(6)
2.6 Markov Chains	28	(1)
2.7 Moments of Record Values	29	(5)
2.7.1 Weibull Distribution	30	(1)
2.7.2 Power Function Distribution	31	(1)
2.7.3 Pareto Distribution	31	(1)
2.7.4 Extreme Value Distribution	32	(1)
2.7.5 Normal Distribution	32	(1)
2.7.6 Covariance and Correlations	33	(1)
2.8 A Discrete Interlude	34	(4)
2.9 Geometric Results	38	(3)
2.10 Counting Process and k-Records	41	(3)
2.10.1 A Point Process View	41	(1)
2.10.2 k-Record Statistics	42	(2)
Exercises	44	(7)
3. Moment Relations, Bounds and Approximations	51	(42)
3.1 Introduction	51	(1)
3.2 Exponential Distribution	52	(2)
3.3 Weibull Distribution	54	(2)
3.4 Gumbel Distribution	56	(5)
3.5 Lomax Distribution	61	(2)
3.6 Normal Distribution	63	(2)
3.7 Logistic Distribution	65	(3)
3.8 Bounds and Approximations	68	(13)
3.9 Results for k-Records	81	(8)
Exercises	89	(4)
4. Characterizations	93	(28)
4.1 Introduction	93	(1)
4.2 Characterizing Properties of Record Values	94	(3)
4.2.1 The Moment Sequence	95	(1)
4.2.2 Regression of Adjacent Record Values	96	(1)
4.3 Families of Distributions	97	(4)
4.3.1 Families Defined by Reliability Properties	97	(2)
4.3.2 Linear Regressions of Adjacent Record Values	99	(2)
4.4 The Exponential Distribution	101	(7)
4.4.1 The Incomplete Catalog	101	(2)
4.4.2 Integrated Cauchy Functional Equation	103	(2)
4.4.3 Characterizations Based on the ICFE and Other Functional Equations	105	(2)
4.4.4 Lower Record Statistics	107	(1)
4.5 Other Continuous Distributions	108	(3)
4.6 Geometric-Tail Distributions	111	(3)
4.7 Dependence Structures of Record Values and Order Statistics	114	(2)
Exercises	116	(5)
5. Inference	121	(62)
5.1 Introduction	121	(1)
5.2 Maximum Likelihood Estimation	122	(5)
5.3 Best Linear Unbiased Estimation	127	(16)
5.4 Best Linear Invariant Estimation	143	(2)
5.5 Interval Estimation and Tests of Hypotheses	145	(5)
5.6 Point Prediction	150	(6)
5.6.1 Best Linear Unbiased Prediction	150	(3)
5.6.2 Best Linear Invariant Prediction	153	(1)
5.6.3 Asymptotic Linear Prediction	154	(2)
5.7 Interval Prediction	156	(7)
5.7.1 Prediction Intervals Based on BLUE's	156	(3)
5.7.2 Conditional Prediction Intervals	159	(2)
5.7.3 Tolerance Region Prediction	161	(1)
5.7.4 Bayesian Prediction Intervals	162	(1)
5.8 Illustrative Examples	163	(6)
5.9 Inference with Records and Inter-Record Times	169	(1)
5.10 Distribution-Free Tests in Time-Series Using Records	170	(5)
Exercises	175	(8)
6. General Record Models	183	(40)
6.1 Introduction	183	(1)
6.2 Geometrically Increasing Populations	184	(3)
6.3 The F(Alpha) Record Model	187	(7)
6.3.1 Finite-sample Properties	188	(2)
6.3.2 Asymptotic Properties	190	(4)
6.4 Linear Drift Record Model	194	(4)
6.5 The Pfeifer Model	198	(4)
6.6 Characterizations	202	(6)
6.6.1 F(Alpha) and Linear Drift Record Models	202	(5)
6.6.2 The Pfeifer Model	207	(1)
6.7 Records From Dependent Sequences	208	(7)
6.7.1 Markov Sequences	209	(2)
6.7.2 Exchangeable Observations	211	(2)
6.7.3 Dependent Models Based on Archimedean Copula	213	(1)
6.7.4 A Random Power Record Model	214	(1)
Exercises	215	(8)
7. Random and Point Process Record Models	223	(42)
7.1 Introduction	223	(1)
7.2 Basic Random Record Model	224	(9)
7.2.1 Joint Distribution of Record Values	224	(5)
7.2.2 Dependence Structure of Record Values and Record Counts	229	(2)
7.2.3 Number of Records	231	(2)
7.3 Basic Point Process Model and Applications	233	(3)
7.4 Records Over Poisson Processes	236	(7)
7.4.1 Homogeneous Poisson Pacing Process	236	(3)
7.4.2 Comparison with the Classical Model	239	(2)
7.4.3 Nonhomogeneous Poisson Pacing Process	241	(2)
7.5 Records Over a Renewal Process	243	(3)
7.6 Records Over Birth Processes	246	(6)
7.6.1 General Birth Process	246	(2)
7.6.2 Yule Process	248	(1)
7.6.3 Poisson Record Arrival Process	249	(3)
7.7 Records Over Other Pacing Processes	252	(2)
7.8 The Secretary Problem	254	(4)
Exercises	258	(7)
8. Higher Dimensional Problems	265	(14)
8.1 How Should We Define Multivariate Records?	265	(2)
8.2 Bivariate Records With Independent Coordinates	267	(4)
8.3 Concomitants of Records	271	(3)
8.4 Lower and Upper Records and the Record Range	274	(3)
8.5 Records in Partially Ordered Sets	277	(1)
Exercises	277	(2)
Bibliography	279	(24)
Author Index	303	(6)
Subject Index	309